General linear summation of the Vilenkin-Fourier series (Q1076926)

This paper is devoted to general linear summation of Vilenkin-Fourier series in \(L^ p[0,1)\) (1\(\leq p\leq \infty)\). It consists of two parts. In the first part the author points out a simple method of how to estimate the rate of convergence for the case \(1\leq p\leq \infty\). Before this S. Yano, A. V. Efimov etc. discussed some special summation in rather complex methods. In the second part, the author discusses the case \(1<p<\infty\), and gives the following result: Let \(f\in L^ p[0,1)\) \((1<p<\infty)\), \(K_ p\in L^ 1[0,1)\), \(K_ p^{{\hat{\;}}}(t)\) a nonincreasing sequence and \(K_ p^{{\hat{\;}}}(k)=1\), \((k<M_{\ell_ 0})\), \(K_ p^{{\hat{\;}}}(j)<1\) \((j\geq M_{\ell_ 0},\ell_ 0\in {\mathbb{P}})\) then \[ \tilde A_ p(\sum^{\infty}_{\ell =\ell_ 0}[a_{\ell}\omega (f,1/M_{\ell +1})]^{s_ 2})^{1\quad /s_ 2}\leq \| f-f*K_ p\|_ p\leq A_ p(\sum^{\infty}_{\ell =\ell_ 0}[a_{\ell}\omega (f,1/M_{\ell \quad})]^{s_ 1})^{1/s_ 1} \] where \(s_ 1=\min (2,p)\), \(s_ 2=\max (2,p)\), \(a_{\ell_ 0}=1-K_ p^{{\hat{\;}}}(M_{\ell_ 0})\), \(a_{\ell}=\sqrt{(1-K_ p^{{\hat{\;}}}(M_{\ell}))^ 2-(1-K_ p^{{\hat{\;}}}(M_{l- 1}))^ 2}\) \((\ell >\ell_ 0)\).